A note on arithmetic-geometric-harmonic mean inequality of several positive operators

Document Type : Research Article


1 Department of Mathematics, Faculty of science, University of Zanjan, Zanjan, Iran.

2 Department of Mathematics, Faculty of science, University of Zanjan, Zanjan, Iran


‎Suppose that $B_1,\cdots,B_m$ are positive operators on a Hilbert space $\mathcal{H}$‎. ‎In this paper we generalize the weighted arithmetic‎, ‎geometric and harmonic means as follows‎:


‎{\mathbf a_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf a_2}(k_1,N';B_1,{\mathbf a_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))=\frac{k_1B_1+\cdots+k_mB_m}{N}\\‎

‎{\mathbf h_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf h_2}(k_1,N';B_1,{\mathbf h_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))=\left(\frac{k_1B_1^{-1}+\cdots+k_mB_m^{-1}}{N}\right)^{-1}\\‎

‎{\mathbf g_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf g_2}(k_1,N';B_1,{\mathbf g_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))‎


‎where $\boldsymbol\kappa=(k_1,\cdots,k_m)‎, ‎N=k_1+\cdots+k_m‎, ‎\boldsymbol\kappa'=(k_2,\cdots‎, ‎k_m)$ and $N'=k_2+\cdots+k_m$‎. ‎We show that the arithmetic-geometric-harmonic mean inequality holds‎. ‎Also we investigate nine property of the geometric mean‎.


Main Subjects

[1] H. Alzer, A proof of the arithmetic mean-geometric mean inequality, Amer. Math. Monthly 103 (1996), no. 7, 585.
[2] T. Ando, Topics on operator inequalities, Lecture Notes (mimeographed), Hokkaido Univ., Sapporo, 1978.
[3] M. Bakherad, R. Lashkaripour and M. Hajmohamadi, Extensions of interpolation between the arithmetic-geometric mean inequality for matrices, J. Inequal. Appl. (2017), Paper No. 209, 10 pp.
[4] Y. Bedrani, F. Kittaneh and M. Sababheh, On the weighted geometric mean of accretive matrices, Ann. Funct. Anal. 12 (2021), no. 1, 16 pp.
[5] P. S. Bullen, D. S. Mitrinović and P. M. Vasić, Means and Their Tnequalities, Reidel Publishing Co., Dordrecht, 1988.
[6] J.-C. Kuang, Applied Inequalities, 2nd edition, Hunan Education Press, Changsha, China, 1993. (Chinese)
[7] H. Lee, Y. Lim and T. Yamazaki, Multi–variable weighted geometric means of positive definite matrices, Linear Algebra Appl. 435 (2011), no. 2, 307–322.
[8] J. Liang and G. Shi, Some means inequalities for positive operators in Hilbert spaces, J. Inequal. Appl. 14 (2017), 1–13.
[9] J. Pečarić, Nejednakosti, Element, Zagreb, 1996.
[10] J. Pečarić, T. Furuta, J. Mićić Hot and Y. Seo, Mond - Pecaeic Method in Operator Inequalities, Element, Zagreb, 2005.
[11] J. Pečarić, F. Qi, V. Šimić and S.-L. Xue, Refinements and extension of ans inequality, III, J. Math. Anal. Appl. 227 (1998), no. 2, 439–448.
[12] J. Pečarić and S. Varošanec, A new proof of the arithmetic mean-the geometric mean inequality, J. Math. Anal. Appl. 215 (1997), no. 2, 577–578.
[13] F. Qi, Generalized weighted mean values with two parameters, Proc. Roy. Soc. London Ser A. 454 (1998), no. 1978, 2723–2732.
[14] M. Raissouli, F. Leazizi and M. Chergui, Arithmetic–geometric–harmonic mean of three positive operators, J. Inequal. Pure Appl. Math. 10 (2009), no. 4, Article 117, 11 pp.
[15] A. Seddik, Operator inequalities related to the arithmetic-geometric mean inequality and characterizations, Adv. Oper. Theory 8 (2023), no. 1, Paper No. 8, 43 pp.